Lincoln D' Carr

1
Quantum Phase Transitions in theFermi-Bose
Hubbard model

• Lincoln D. Carr
• JILA
• Physics Department, University of Colorado and
• National Institute for Standards and Technology
• in collaboration with
• Murray Holland

2
Motivation Experiments with Cold Atoms on
Lattices

• Exp. realization of Hubbard model
• Parameters m, J, V
• Use of a Feshbach resonance with fermions in the
  continuum has led to new states of quantum matter
• Strongly interacting Fermi superfluid,
  Breached-pair superfluidity, etc.
• Problem we address What happens to cold fermions
  subject to a Feshbach resonance on a lattice?

3
What are Quantum Phase Transitions in the Hubbard
model?

• Hubbard Hamiltonian e.g. bosons
• E.g. cold bosons in a lattice at T0
• Shallow lattice Superfluid phase
• Atoms flow freely
• Phase correlated between sites
• Deep lattice Mott phase
• Atoms confined to individual sites
• Phase uncorrelated between sites

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Our Fermi-Bose Hubbard Hamiltonian

Fermi-pair boson interconversion
Total number of atoms
Chemical equilibrium
5
Solve for lowest band, deep lattice

• Consider only paired fermions (Jf ltlt Vf)
• Second order degenerate perturbation theory for
  Hf
• Hf maps onto new Hamiltonian
• Spin operators
• Compare to Heisenberg spin Hamiltonian (XXZ model)

6
Two State Approximation and the Phase Diagram

• Same variational wavefunction for each site
• Variational parameters Number angle q and
  Mixing angle c
• Extremize energy E(q,c).
• Mott insulator superfluid boundary appears as a
  saddle point in q.

Fermi-pair/Bose vacuum
One boson, zero Fermi-pairs
Zero bosons, one Fermi-pair
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T0 Phase diagram through crossover
What is a first experimental observable for cold
fermions on a lattice with a Feshbach resonance?
Phase coherence

• (a) n -10 (b) n -1(c) n 1/2(d) n
  10NB g1

Mostly bosons
Mostly Fermi-pairs
8
Field occupation and avoided crossing

• (a) Mixing anglefor field occupation
• Blue ground state
• Redmaximum
• (b) Avoided crossing in phasediagram

Bosons
Fermi-pairs
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Conclusion

• Proposed general Fermi-Bose Hubbard model which
  describes BCS-BEC crossover on a lattice
• Recovers Fermi and Bose Hubbard phase
  diagramsfor large positive and negative detuning
  n
• Solved zero to two fermions per site
• Experimental case in 3D 106 fermions, 1003
  sites
• Both broad and narrow Feshbach resonances
• Predicted clear observable in an experiment
• Phase coherence
• Mott phases consist of dressed Fermi-pairs/bosons

? E-print cond-mat/0501156